For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

learn more… | top users | synonyms (5)

16
votes
2answers
726 views

How come such different methods result in the same number, $e$?

I guess the proof of the identity $$ \sum_{n = 0}^{\infty} \frac{1}{n!} \equiv \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x $$ explains the connection between such different calculations. How ...
10
votes
2answers
1k views

If the series $\sum_0^\infty a_n$ converges, then so does $\sum_1^\infty \frac{\sqrt{a_n}}{n} $

Problem: Suppose that for every $n\in\mathbb{N}$, $a_n\in\mathbb{R}$ and $a_n\ge 0$. Given that $$\sum_0^\infty a_n$$ converges, show that $$\sum_1^\infty \frac{\sqrt{a_n}}{n} $$ ...
7
votes
5answers
517 views

Limit of $s_n = \int\limits_0^1 \frac{nx^{n-1}}{1+x} dx$ as $n \to \infty$

Let $s_n$ be a sequence defined as given below for $n \geq 1$. Then find out $\lim\limits_{n \to \infty} s_n$. \begin{align} s_n = \int\limits_0^1 \frac{nx^{n-1}}{1+x} dx \end{align} I have ...
5
votes
10answers
514 views

Why does $\sum_{n = 0}^\infty \frac{n}{2^n}$ converge to 2? [duplicate]

Apparently, $$ \sum_{n = 0}^\infty \frac{n}{2^n} $$ converges to 2. I'm trying to figure out why. I've tried viewing it as a geometric series, but it's not quite a geometric series since the ...
4
votes
1answer
351 views

Find $\lim\limits_{n \to \infty} \frac{1}{n}\sum\limits^{2n}_{r =1} \frac{r}{\sqrt{n^2+r^2}}$

Find $$\lim_{n \to \infty} \frac{1}{n}\sum^{2n}_{r =1} \frac{r}{\sqrt{n^2+r^2}}$$ My approach : $$\lim_{n \to \infty} \frac{1}{n}\sum^{2n}_{r =1} \frac{r}{\sqrt{n^2+r^2}} =\lim_{n \to \infty} \frac{...
67
votes
4answers
9k views

Motivation for Ramanujan's mysterious $\pi$ formula

The following formula for $\pi$ was discovered by Ramanujan: $$\frac1{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!$$ Does anyone know how it works, or ...
36
votes
4answers
1k views

Infinite Series $\sum_{n=1}^\infty\frac{H_n}{n^32^n}$

I'm trying to find a closed form for the following sum $$\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n},$$ where $H_n=\displaystyle\sum_{k=1}^n\frac{1}{k}$ is a harmonic number. Could you help me with it?
30
votes
3answers
753 views

Infinite Series $\sum_{n=1}^\infty\frac{H_n}{n^22^n}$

How can I prove that $$\sum_{n=1}^{\infty}\frac{H_n}{n^2 2^n}=\zeta(3)-\frac{1}{2}\log(2)\zeta(2).$$ Can anyone help me please?
42
votes
4answers
1k views

Proving $\sum\limits_{l=1}^n \sum\limits _{k=1}^{n-1}\tan \frac {lk\pi }{2n+1}\tan \frac {l(k+1)\pi }{2n+1}=0$

Prove that $$\sum _{l=1}^{n}\sum _{k=1}^{n-1}\tan \frac {lk\pi } {2n+1}\tan \frac {l( k+1) \pi } {2n+1}=0$$ It is very easy to prove this identity for each fixed $n$ . For example let $n = 6$; ...
24
votes
4answers
824 views

Closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$.

What is the closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$? We can use Fourier series of $e^{-bx}$ ($|x|<\pi$) to evaluate $\sum_{n=-\infty}^{\infty}\frac{1}{n^2+b^2}$. But this ...
17
votes
5answers
1k views

Evaluating the infinite product $\prod_{k=2}^{\infty }\left ( 1-\frac{1}{k^2} \right ).$ [closed]

Evaluate $$\lim_{ n\rightarrow\infty }\prod_{k=2}^{n}\left ( 1-\frac{1}{k^2} \right ).$$ I can't see anything in this limit , so help me please.
14
votes
7answers
23k views

Why is $\arctan(x)=x-x^3/3+x^5/5-x^7/7+\dots$?

Why is $\arctan(x)=x-x^3/3+x^5/5-x^7/7+\dots$? Can someone point me to a proof, or explain if it's a simple answer? What I'm looking for is the point where it becomes understood that trigonometric ...
32
votes
3answers
14k views

$\sum k! = 1! +2! +3! + \cdots + n!$ ,is there a generic formula for this?

I came across a question where I needed to find the sum of the factorials of the first $n$ numbers. So I was wondering if there is any generic formula for this? Like there is a generic formula for ...
17
votes
2answers
2k views

Series of logarithms $\sum\limits_{k=1}^\infty \ln(k)$ (Ramanujan summation?)

I had this question earlier, so to say as a "standalone" problem, but now it pops up in context of an analysis with the lngamma-function. As well as we can convert the question of sums of like powers $...
9
votes
3answers
592 views

A series to prove $\frac{22}{7}-\pi>0$

After T. Piezas answered Is there a series to show $22\pi^4>2143\,$? a natural question is Is there a series that proves $\frac{22}{7}-\pi>0$? One such series may be found combining ...
7
votes
3answers
6k views

Properties of $\liminf$ and $\limsup$ of sum of sequences

Let $\{s_n\}$ and $\{t_n\}$ be sequences. I've noticed this inequality in a few analysis textbooks that I have come across, so I've started to think this can't be a typo: $\limsup\limits_{n \...
10
votes
4answers
311 views

Infinite Series $\sum_{n=1}^{\infty}\frac{1}{\prod_{k=1}^{m}(n+k)}$

How to prove the following equality? $$\sum_{n=1}^{\infty}\frac{1}{\prod_{k=1}^{m}(n+k)}=\frac{1}{(m-1)m!}.$$
5
votes
2answers
469 views

If $ x_n \to x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} \to x $

I have this problem I'm working on. Hints are much appreciated (I don't want complete proof): In a normed vector space, if $ x_n \longrightarrow x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} \...
4
votes
3answers
14k views

Proving formula for product of first n odd numbers

I have this formula which seems to work for the product of the first n odd numbers (I have tested it for all numbers from $1$ to $100$): $$\prod_{i = 1}^{n} (2i - 1) = \frac{(2n)!}{2^{n} n!}$$ How ...
3
votes
3answers
404 views

Cauchy Sequence. What is this question actually telling me?

Let $(a_n)$ be a sequence such that $\lim\limits_{N\to\infty} \sum_{n=1}^n |a_n-a_{n+1}|<\infty$. Show that $(a_n)$ is Cauchy. So basically I am told that the sum of the difference isn't ...
5
votes
5answers
2k views

The convergence of $\sqrt {2+\sqrt {2+\sqrt {2+\ldots}}}$ [duplicate]

I would like to know if this sequence converges $\sqrt {2+\sqrt {2+\sqrt {2+\ldots}}}$. I know this sequence is increasing monotone, but I couldn't prove it's bounded. Thanks
5
votes
2answers
830 views

Another Epsilon-N Limit Proof Question

How to prove the limit of the following sequence using epsilon-N argument. $$a_n=\frac{3n^2+2n+1}{2n^2+n}$$ I took the limit to be $\frac{3}{2}$ and proceeded with the argument, $$\left|\frac{3n^2+...
1
vote
1answer
255 views

Series and integrals for inequalities and approximations to $\pi$

Fundamentals Two beautiful expressions that relate $\pi$ to its convergents are Dalzell integral $$\frac{22}{7}-\pi=\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx$$ (see Why do we need an integral to prove ...
39
votes
1answer
3k views

Convergence of $\sum \limits_{n=1}^{\infty}\sin(n^k)/n$

Does $S_k= \sum \limits_{n=1}^{\infty}\sin(n^k)/n$ converge for all $k>0$? Motivation: I recently learned that $S_1$ converges. I think $S_2$ converges by the integral test. Was the question ...
32
votes
6answers
2k views

A closed form for the sum $\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$

How can I find a closed form for the following sum? $$\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$$ ($H_n=\sum_{k=1}^n\frac{1}{k}$).
29
votes
10answers
4k views

Direct Proof that $1 + 3 + 5 + \cdots+ (2n - 1) = n\cdot n$

Prove that for all integers $n$, $n \geq 1$, $$1 + 3 + 5 + \cdots + (2n - 1) = n\cdot n$$ How would I go about proving this?
23
votes
4answers
908 views

How close can $\sum_{k=1}^n \sqrt{k}$ be to an integer?

How close can $S(n) = \sum_{k=1}^n \sqrt{k}$ be to an integer? Is there some $f(n)$ such that, if $I(x)$ is the closest integer to $x$, then $|S(n)-I(S(n))|\ge f(n)$ (such as $1/n^2$, $e^{-n}$, ...). ...
18
votes
3answers
1k views

How to prove Chebyshev's result: $\sum_{p\leq n} \frac{\log p}{p} \sim\log n $ as $n\to\infty$?

I saw reference to this result of Chebyshev's: $$\sum_{p\leq n} \frac{\log p}{p} \sim \log n \text{ as }n \to \infty,$$ and its relation to the Prime Number Theorem. I'm looking into an information-...
16
votes
4answers
1k views

Where do summation formulas come from?

It's a classic problem in an introductory proof course to prove that $\sum_{ i \mathop =1}^ni = \frac{n(n+1)}{2}$ by induction. The problem with induction is that you can't prove what the sum is ...
14
votes
5answers
1k views

Find the infinite sum of the series $\sum_{n=1}^\infty \frac{1}{n^2 +1}$

This is a homework question whereby I am supposed to evaluate: $$\sum_{n=1}^\infty \frac{1}{n^2 +1}$$ Wolfram Alpha outputs the answer as $$\frac{1}{2}(\pi \coth(\pi) - 1)$$ But I have no idea ...
9
votes
1answer
1k views

Existence of a limit associated to an almost subadditive sequence

I want to prove that a sequence lives in a specific interval; I did prove that lives in a bigger interval, but not in the one I want. Let $ a_n $ a sequence such that for any n,m $$a_n + a_m -1 &...
14
votes
2answers
791 views

Convergence of $\sum_{n=1}^\infty \frac{\sin^2(n)}{n}$

Does the series $$ \sum_{n=1}^\infty \frac{\sin^2(n)}{n} $$ converge? I've tried to apply some tests, and I don't know how to bound the general term, so I must have missed something. Thanks in ...
12
votes
5answers
6k views

Is there any formula for the series $1 + \frac12 + \frac13 + \cdots + \frac 1 n = ?$

Is there any formula for this series? $$1 + \frac12 + \frac13 + \cdots + \frac 1 n .$$
11
votes
9answers
8k views

I have to show $(1+\frac1n)^n$ is monotonically increasing sequence

I have to show $(1+\frac1n)^n$ is monotonically increasing sequence. $U_n=(1+\frac1n)^n$, $U_{n+1}=(1+\frac1{n+1})^{n+1}$ I must show $U_{n+1}-U_n\geq0$ i.e. to show $(1+\frac1{n+1})^{n+1}-(1+\...
17
votes
1answer
878 views

Is there a constructive proof of this characterization of $\ell^2$?

I would like to revisit this question, which can be equivalently stated as: Proposition. Let $(a_n)$ be a sequence of real (or complex) numbers such that $\sum a_n b_n$ converges for every $(b_n) ...
9
votes
4answers
824 views

Why does this expected value simplify as shown?

I was reading about the german tank problem and they say that in a sample of size $k$, from a population of integers from $1,\ldots,N$ the probability that the sample maximum equals $m$ is: $$\frac{\...
7
votes
2answers
8k views

If a series converges, then the sequence of terms converges to $0$.

Following the guidelines suggested in this meta discussion, I am going to post a proposed proof as an answer to the theorem below. I believe the proof works, but would appreciate any needed ...
8
votes
2answers
405 views

recurrence relation $f(n)=5f(n/2)-6f(n/4) + n$

I've been trying to solve this recurrence relation for a week, but I haven't come up with a solution. $f(n)=5f(n/2)-6f(n/4) + n$ Solve this recurrence relation for $f(1)=2$ and $f(2)=1$ At first ...
6
votes
1answer
223 views

How to prove $\prod_{i=1}^{\infty} (1-a_n) = 0$ iff $\sum_{i=1}^{\infty} a_n = \infty$?

Given ${a_n}$ is infinite sequence, and $0 < a_n < 1$, how to prove $$\prod_{i=1}^{\infty} (1-a_n) = 0 \text{ if and only if } \sum_{i=1}^{\infty} a_n = \infty$$ Thanks for your help.
8
votes
2answers
4k views

Does $\sum\limits_{k=1}^n 1 / k ^ 2$ converge when $n\rightarrow\infty$?

I can prove this sum has a constant upper bound like this: $$\sum_{k=1}^n \frac1{k ^ 2} \lt 1 + \sum_{k=2}^n \frac 1 {k (k - 1)} = 2 - \frac 1 n \lt 2$$ And computer calculation shows that sum ...
7
votes
2answers
247 views

Checking my understanding: $1 - 1 + 1 - 1 + 1 - … = \frac{1}{2}$

I've recently run into a proof that claims that $1 - 1 + 1 - 1 + 1 - 1 ... = \frac{1}{2}$ that proceeds as follows: Let $S = 1 - 1 + 1 - 1 + 1 - 1 + ...$. Then $$S = 1 - (1 - 1 + 1 - 1 + 1 - 1 ...)...
4
votes
6answers
285 views

Find the sum of the series $\sum \frac{1}{n(n+1)(n+2)}$

I got this question in my maths paper Test the condition for convergence of $$\sum_{n=1}^\infty \frac{1}{n(n+1)(n+2)}$$ and find the sum if it exists. I managed to show that the series ...
3
votes
4answers
1k views

Formula for calculating $\sum_{n=0}^{m}nr^n$

I want to know the general formula for $\sum_{n=0}^{m}nr^n$ for some constant r and how it is derived. For example, when r = 2, the formula is given by: $\sum_{n=0}^{m}n2^n = 2(m2^m - 2^m +1)$ ...
8
votes
5answers
2k views

Geometric mean never exceeds arithmetic mean

This was a mathematical induction question proposed in a textbook, and I've exhausted multiple approaches (proving RHS - LHS > 0, splitting the fraction, fractional exponents, etc.) The geometric ...
6
votes
3answers
194 views

Compute the limit $\lim_{n \to \infty} \frac{n!}{n^n}$

I am trying to calculate the following limit without Stirling's relation. \begin{equation} \lim_{n \to \infty} \dfrac{n!}{n^n} \end{equation} I tried every trick I know but nothing works. Thank you ...
2
votes
5answers
597 views

Finding generating function for the recurrence $a_0 = 1$, $a_n = {n \choose 2} + 3a_{n - 1}$

I am trying to find generating function for the recurrence: $a_0 = 1$, $a_n = {n \choose 2} + 3a_{n - 1}$ for every $n \ge 1$. It looks like this: $a_0 = 1$ $a_1 = {1 \choose 2} + 3$ $a_2 = {2 \...
1
vote
1answer
662 views

Taylor series for $\log(1+x)$ and its convergence

I know the taylor series of $\log(1+x)$. However I don't understand how to find the convergence for $x>1$ and divergence if $x<1$.
43
votes
3answers
2k views

Does the series $ \sum\limits_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}} $ converge or diverge?

Does the following series converge or diverge? I would like to see a demonstration. $$ \sum_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}}. $$ I can see that: $$ \sum_{n=1}^{\infty} \frac{1}{n^{1 + |\...
37
votes
6answers
1k views

A series involving the harmonic numbers : $\sum_{n=1}^{\infty}\frac{H_n}{n^3}$

Let $H_{n}$ be the nth harmonic number defined by $ H_{n} = \sum_{n=1}^{n} \frac{1}{k}$. I'm interested in knowing how to show that $$\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}.$$ I tried ...
22
votes
3answers
617 views

Closed form for $\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n}$

Please help me to find a closed form for the sum $$\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n},$$ where $H_n$ are harmonic numbers: $$H_n=\sum_{k=1}^n\frac{1}{k}=\frac{\Gamma'(n+1)}{n!}+\gamma.$$